Some useful probability distributions

In practice in statistics we often use a set of well-known distributions. These have specific mathematical forms that come with parameters to make them flexibly useful.

Here are some of the most useful and commonly-used ones:

Name	Expression and R function	Parameters	Explanation
Binomial	${n \choose x}p^x(1-p)^{(n-x)}$ (dbinom() in R)	Number of 'trials' $n$ 'Success probability' $p$	How many 'successes' from $n$ trials? E.g. 'alternate' alleles in a sample of $n$ chromosomes? $x\in 0, \cdots, n$
Beta	$\frac{1}{B(\alpha,\beta)} x^{\alpha-1} (1-x)^{\beta-1}$ (dbeta() in R)	'Shape' parameters $\alpha$ and $\beta$	Modelling proportions or frequencies E.g. allele frequencies or cell proportions $x \in [0,1]$
Poisson	$\frac{\lambda^xe^-\lambda}{k!}$ (dnorm() in R)	Mean or rate parameter $\lambda$	How often a (rare) event occurs in a given time interval $x\in\{0, 1, 2, ...\}$
Normal or Gaussian	$\frac{1}{\sqrt{2\pi v}} e^{-\frac{1}{2}\frac{(x-\mu)^2}{v}}$ (dnorm() in R)	Mean $\mu$ Variance $v$	Ubiquitously useful as we'll see later $x \in \text{Real numbers}$

If you don't understand the maths above, don't worry. You can understand these distributions by plotting what they look like as we'll do below.

Normalising constants

Many of these mathematical expressions have complicated-looking bit at the front that doesn't depend on $x$. For example - the normal distribution has this bit:

$$\frac{1}{\sqrt{2\pi v}}$$

while the beta distribution has this bit:

$$\frac{1}{B(\alpha,\beta)}$$

(here $B()$ is the 'beta function').

This bits can look complicated but the don't depend on $x$. In fact, they are just normalising constants: their purpose is to ensure the distribution sums to $1$ over all the possible values of $x$.

Question. The expression $n \choose x$ in front of the binomial isn't a normalising constant in the same way - why not?

Distributions challenge

Go and explore some distributions now - and figure out how to answer these questions:

Challenge questions

• Where would I find 95% of the mass of a normal distribution with mean 10 and variance 20?

• Suppose I sample 100 chromosomes from a population, to examine a variant of interest that is at 1% frequency. What's the chance I will detect at least one chromosome with the allele? (Hint this is a job for the binomial distribution).

• Can you generate a beta distribution that looks like a one-sided smile? (High one one side, low on the other.)